In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
Vo
l
.
1
0
, No
.
2
, Ju
n
e
20
1
9
, p
p
.
1
1
2
3
~
1
132
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
2.pp1123-1132
1
1
23
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
Starting torque an
d torque r
ipple reduction using SV
P
WM
based ve
ctor control of
indu
ction motor with
nin
e-level
cascaded multilevel In
v
erter
fed w
ith sol
a
r PV pow
er
Sh
a
s
h
i
b
h
us
ha
n
1
, S
avita S
on
ol
i
2
1
Depart
emen
t
o
f
E
CE, Si
r M
V
IT, India
2
D
e
part
emen
t of ECE
, RYM En
g
i
n
eerin
g Coll
e
g
e, In
d
i
a
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
c
e
i
v
e
d
No
v
2
7
,
2
018
Re
vise
d Jan
2
9
, 2019
Ac
ce
p
t
ed
M
ar 1
4
,
2
019
Th
is
p
ap
er
i
s
an
a
t
t
em
p
t
t
o
dev
e
lo
p
an
I
nd
uction
M
o
t
o
r
D
r
iv
e
Sy
ste
m
w
ith
Multilevel
Inver
t
er
t
opology
f
o
r
reduced
t
orque
ripp
l
e
a
pplicat
i
o
n
.
A
N
i
n
e
lev
e
l-cascad
ed
m
ultil
e
vel
i
n
v
e
rter
i
s
dev
e
lo
ped
fo
r
t
h
e
ind
u
c
t
i
o
n
m
ot
or
d
riv
e
with
S
VPW
M
con
trol
p
owered
b
y
b
o
o
s
t con
v
erter f
e
d
u
s
i
ng so
la
r
P
V
s
u
p
p
ly
.
The
SVP
W
M
c
o
n
t
ro
l
ba
se
d
im
ple
m
e
n
ta
t
i
o
n
o
f
ve
c
t
or
c
on
tr
ol
u
sing
a
m
u
lti
l
evel
i
nverter
t
o
p
o
l
o
gy
need
s
a
mult
il
evel
S
VPWM
c
ont
ro
l
t
ech
ni
qu
e,
which
i
s
i
mplemented
i
n
this
p
ap
er.
T
h
e
S
o
lar
p
o
w
e
r
su
pp
li
ed
i
s
a
ppl
ied
w
ith
th
e
M
P
P
T
t
ech
ni
qu
e
and
t
h
e
supp
lied
D
C
p
o
w
er
i
s
f
e
d
t
o
t
h
e
t
hre
e
phas
e
cascad
ed
9
l
evel
m
ulti
lev
e
l
i
n
v
e
rter.
Th
e
vect
or
c
on
trol
o
f
i
n
d
uc
t
i
o
n
m
oto
r
i
s
carried
out
u
s
i
ng
th
e
S
V
PWM
t
echn
i
qu
e
on
t
he
m
u
ltil
e
v
e
l
t
opolo
g
y.
T
he
to
rqu
e
r
ip
ple
re
duct
i
o
n
i
n
t
h
e
out
pu
t
i
s
o
b
s
erved
and
com
p
a
r
ed
wi
th
t
he
vect
or
c
on
trol
o
f
i
n
d
u
ctio
n
moto
r.
M
a
tlab
base
d
impl
e
m
entation
is
carried
ou
t an
d t
h
e
res
u
lts are tab
ul
ated
a
nd
inf
erred
.
K
eyw
ord
s
:
Mu
lt
ile
ve
l in
ve
r
t
e
r
Mu
lt
ile
ve
l spa
ce
vector
p
ulse
wi
dt
h
mo
dul
at
io
n
Torq
ue
r
ip
pl
e
r
e
duc
t
i
o
n
V
e
c
t
or
c
o
n
tro
l
o
f i
n
duc
t
i
o
n
motor
Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Sh
as
hi
bhu
sh
an,
Depa
rtem
ent o
f
E
CE,
Sir
MVIT
,
Interna
tio
na
l
A
i
rp
ort R
o
ad,
H
una
sam
a
r
a
nah
a
lli,
B
en
ga
lur
u
, K
ar
na
t
a
k
a
562
157
, In
d
i
a
Em
ail:
sha
s
hig
a
dig
i
1
2
3
4
@
g
m
a
i
l.
com
1.
I
N
TR
OD
U
C
TI
O
N
H
i
g
h
er
E
lec
t
ro
m
a
gne
tic
c
om
pat
i
b
i
l
i
t
y
i
n
M
ul
t
i
le
ve
l
In
ver
t
e
r
a
nd
e
xt
en
si
v
e
u
se
o
f
Induc
t
i
on
m
o
t
o
r
s
in
t
he
i
nd
us
t
r
ie
s
has
bro
u
g
h
t
i
n
a
h
i
g
h
er
d
em
and
of
t
he
m
ulti
le
ve
l
i
n
v
e
rter
b
ase
d
i
n
duc
t
i
o
n
m
o
t
or
d
ri
ves.
S
o
lar
ba
se
d
s
u
pp
l
y
f
or
t
hese
i
n
duc
t
i
on
m
ot
or
d
ri
ves
w
o
ul
d
inc
r
e
a
s
e
the
cos
t
e
ffe
c
t
i
v
en
ess
of
t
he
i
n
d
u
c
tio
n
motor
dr
i
v
e
sy
stem
s
sinc
e
t
h
e
pow
er
c
har
g
es
a
re
goin
g
t
o
be
n
ul
lifie
d
e
x
ce
pt
f
or
t
he
capi
t
a
l
c
ha
rge
s
.
A
F
i
ve
Le
v
e
l
In
v
e
r
t
er
c
o
n
t
r
ol
l
e
d
by
c
a
rri
er
b
as
ed
S
PW
M
t
echni
qu
e
i
s
u
s
e
d
w
i
t
h
o
pe
n
en
d
w
i
nd
i
ng
1H
P
in
d
u
ct
io
n
m
o
t
o
r
[
1
]
.
T
w
o
t
h
r
e
e
l
e
v
e
l
i
n
v
e
r
t
e
r
e
a
c
h
c
o
m
p
r
i
s
i
n
g
t
w
o
2
l
e
v
e
l
i
n
v
er
te
rs
a
r
e
p
lac
e
d
e
ach
a
t
o
n
e
e
n
d
o
f
t
he
ope
n-e
n
de
d i
n
duc
t
i
o
n
m
o
t
or
. In [2] the
num
be
r of sw
i
tc
hes
are
r
e
duce
d
b
ut incre
as
i
ng t
h
e n
u
m
b
er
of
l
e
vel
s
t
o
six.
A
s
ix
l
e
v
e
l
o
u
t
pu
t
is
o
b
t
a
i
ne
d
b
y
u
s
i
ng
a
c
ombi
na
ti
o
n
o
f
a
t
hre
e
l
ev
el
i
nve
r
t
er
w
it
h
t
h
e
tw
o
l
e
ve
l
in
v
e
r
t
e
r
on
e
i
t
he
r
side
o
f
the
ope
n-en
de
d
i
n
duc
t
i
o
n
m
otor.
Th
is
t
o
p
o
l
og
y
n
e
e
d
s
thr
ee
is
o
l
ate
d
p
ower
s
u
p
p
l
i
e
s
a
s
t
her
e
are
thre
e tw
o
level i
n
ver
t
e
r
s c
o
mpr
i
se
d
in i
t.
A
prop
osa
l
o
f
a
struc
t
ure
f
o
r
mult
ile
v
e
l
i
n
ve
r
t
e
r
t
op
ol
o
g
ie
s
fo
r
s
t
and
alone
P
V
syst
em
i
s
seen
i
n
[3]
.
The
m
u
lti-w
i
n
d
i
ng
to
p
o
lo
gy
i
n
v
erter
gi
ves
bet
t
er
r
e
s
u
lts
c
om
par
e
d
t
o
oth
e
r
t
y
pe
s
of
m
u
l
t
i
le
ve
l
in
verter
s.
T
he
lite
rat
u
re has p
resente
d
the
si
n
g
l
e pha
se PI con
t
ro
l
l
er base
d
m
u
l
t
i
l
e
v
e
l
i
nve
rt
er
f
o
r
g
ri
d
co
nn
e
c
t
i
on
w
i
t
h
l
e
ss
er
TH
D
[4].
I
t
ha
s
bee
n
p
rese
nt
e
d
a
h
e
x
a
g
ona
l
a
nd
1
2
-
si
de
d
po
l
y
g
o
nal
v
o
l
t
a
ge
s
pac
e
v
ec
t
o
r
w
ith
c
a
s
ca
d
e
d
tw
o-
leve
l
w
i
t
h
i
n
duc
t
i
on
m
o
t
o
r
dri
v
e
i
n
[
5]
.
The
TH
D
va
lue
i
s
m
i
n
im
ize
d
i
n
thi
s
l
ite
rat
u
re
a
ls
o.
I
t
ha
s
bee
n
ntr
o
d
u
ce
d
hy
b
r
id
m
u
l
t
i
le
ve
l
in
verter
t
o
p
o
l
ogy
f
or
o
pe
n-
end
w
i
nd
in
g
i
n
d
u
c
ti
on
m
ac
hin
e
u
sing
t
wo
l
e
v
el
inverter
in ser
i
es with a
c
apacit
or-fed H-
bridge cell,
w
hi
c
h
el
im
ina
t
es 1
8 c
l
am
pin
g
d
io
de
s
in [6]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
In
t
J Po
w El
ec
&
Dri
S
y
s
t
,
Vo
l
. 1
0
,
No
.
2
, Ju
n
e
2
019
:
1
123 –
1
1
32
1
124
A
lit
e
r
a
t
ure
su
rve
y
o
n
mul
t
ile
vel
in
ver
t
e
r
t
op
o
l
o
g
i
es
a
n
d
it
c
o
n
t
r
o
l
stra
t
e
gie
s
a
re
p
r
e
sente
d
i
n
[1]
.
Ma
in
ly
t
o
p
o
l
o
g
ie
s
l
i
k
e
d
i
ode
c
lam
p
e
d
m
ult
i
leve
l
i
n
v
e
rter
,
ca
pac
i
t
or
c
lam
p
ed
a
nd
c
a
sca
d
ed
i
s
c
o
nsi
d
er
ed
a
n
d
some
P
WM
t
e
c
hn
i
que
s
als
o
.
This
i
s
he
lp
ful
in
a
nal
y
z
i
n
g
d
iffere
n
t
t
y
p
e
s
of
t
opo
l
o
g
i
es
a
nd
c
o
n
t
r
o
ls.
The
ca
sc
ade
d
m
u
lti
leve
l
gr
id
c
onn
e
c
ted
i
n
verter
s
ha
s
ener
gy
ba
l
a
nce
p
r
o
b
l
ems.
A
l
ev
el
s
hi
ft
ed
a
n
d
p
h
a
se
-shi
ft
ed
P
W
M
s
w
it
h
e
n
er
gy
ba
lance
is
s
t
u
d
i
e
d
i
n
[7].
I
n
2
0
14
m
o
du
lar
m
u
l
tile
ve
l
-
casc
a
de
d
in
ver
t
e
r
d
ri
ve
n
in
d
u
c
tio
n
motor
w
i
th
o
u
t
se
nsor
i
s
pro
p
o
se
d
[8]
.
D
eve
l
ope
d
D
u
al
i
n
v
e
rter
f
ed
o
pen-
e
nd
s
t
a
t
or
w
in
di
n
g
i
nd
uc
t
i
o
n
motor
dri
v
e
w
i
th
o
p
t
im
al
p
u
l
se
w
id
t
h
m
o
d
u
l
a
t
io
n.
I
t
is
d
isc
u
sse
d
t
h
a
t
d
is
t
r
ibu
t
e
d
M
P
P
T
for
gri
d
c
on
nec
t
e
d
m
od
u
l
ar
c
a
s
c
a
d
ed
H-B
r
i
dg
e
mu
lti
l
e
v
el PV i
n
v
e
rt
er i
s ap
pl
i
e
d
[9
].
I
n
t
h
i
s
paper
9
-
l
evel
m
u
lti
le
vel
i
nvert
er
i
s
pro
pose
d
w
i
t
h
S
V
P
W
M
c
ont
ro
l
f
o
r
redu
c
tio
n
of
s
t
a
rt
i
n
g
tor
que
a
nd
tor
q
u
e
r
i
p
pl
e
re
d
u
ct
ion.
T
his
a
l
so
p
r
o
d
u
ces
f
a
s
t
res
p
o
n
s
e
o
f
s
peed.
T
h
e
c
u
rre
nt
r
ip
ple
is
a
l
s
o
reduc
e
d
a
n
d
T
H
D
r
e
duct
i
o
n
a
s
w
e
ll
at
m
otor
t
e
r
m
i
na
ls.
Th
is
p
a
p
e
r
is
o
r
g
anized
a
s
f
o
ll
ows,
S
e
c
t
i
on
II
di
sc
usse
s
t
h
e
Ca
sc
a
d
ed
M
u
l
t
i
le
ve
l
I
n
verter
a
nd
i
t
s
s
w
i
tch
i
ng
s
e
que
nce
,
S
ecti
o
n
III
b
riefs
ab
o
u
t
t
h
e
SV
PWM
con
t
ro
l
o
f
t
he
9
l
e
v
e
l
m
ult
ile
vel
i
nvert
er
,
S
ecti
o
n
IV
d
isc
u
sse
s
t
h
e
pro
p
o
se
d
me
thod
i
n
d
e
t
a
il,
a
n
d
S
ec
t
i
o
n
V
di
sc
usse
s the
resu
lts
f
o
llow
e
d
by
Conc
l
u
si
o
n
and
Re
fer
e
nce
s
.
2.
CASCADE
D
MU
L
T
ILEVEL INV
E
RTER
The
r
e
is
a
g
ro
w
i
n
g
i
n
t
e
r
es
t
i
n
m
ul
t
i
l
e
ve
l
t
o
pol
og
ies
s
i
nc
e
t
h
e
y
c
a
n
e
x
t
en
d
t
h
e
a
p
plica
t
i
on
of
pow
er
elec
tr
on
i
c
s
s
y
stem
s
to
h
ig
h
e
r
volta
ge
s
a
nd
pow
e
r
r
ati
o
s.
M
ul
ti
l
e
v
el
i
nv
ert
e
rs
a
re
t
h
e
m
o
s
t
at
trac
t
i
v
e
tech
n
o
l
o
gy
for
the
m
e
d
i
um
t
o
h
i
g
h
v
o
l
tage
r
a
nge,
w
h
ich
inc
l
ude
s
m
ot
or
d
r
i
v
e
s,
p
ow
er
d
is
t
r
ib
ut
i
on,
pow
er
qua
l
ity a
n
d
po
w
e
r
cond
it
io
n
i
ng ap
p
lica
t
io
ns
. The
ge
nera
l func
t
i
o
n
o
f t
h
is m
ult
ile
ve
l i
n
ve
rter
is to
s
yn
t
h
esize a
des
i
red
vo
l
t
age
from
se
vera
l
se
para
t
e
D
C
so
urc
e
s,
w
hi
c
h
m
a
y
b
e
ob
t
a
ine
d
f
ro
m
b
a
t
t
e
ries
,
f
u
el
cel
l
s
,
or
s
o
l
ar
c
e
l
l
s.
A
p
a
r
ti
cu
l
a
r
ad
va
nt
ag
e
o
f
t
hi
s
t
o
po
log
y
i
s
th
a
t
t
h
e
m
o
d
u
la
t
i
o
n
,
co
ntr
o
l
a
n
d
pro
t
e
c
t
i
o
n
r
e
q
u
i
r
e
ment
s
o
f
ea
ch
b
ri
d
g
e
a
r
e
m
odu
lar.
T
h
e
c
asca
de
d
i
nve
rter
h
as
b
ee
n
l
a
rge
l
y
s
t
udie
d
a
nd
use
d
i
n
t
h
e
var
i
o
u
s
fie
l
d
s
s
uc
h
as dr
i
ve
s,
t
r
a
ns
missi
on
s
yst
e
m
and pow
e
r
c
on
dit
i
o
n
i
ng.
Sp
ec
i
a
l
Fe
at
u
r
e
s
o
f
C
a
sc
ad
ed M
ul
t
i
l
e
v
e
l
In
ve
rt
er
a.
The
se
ries
s
tru
c
ture
a
l
l
o
w
s
a
s
c
a
la
ble,
m
od
u
l
ar
i
z
e
d
c
ircu
it
lay
o
u
t
a
nd
pac
k
ag
i
ng
s
i
nc
e
e
ach
b
ri
dg
e
has
the sa
me
struct
u
re
.
b.
Re
qu
i
r
es
t
he
l
e
a
s
t
n
u
m
b
er
o
f
com
p
o
n
e
n
t
s
a
mong
a
l
l
m
u
l
t
i
l
eve
l
c
o
n
v
erter
s
t
o
a
c
h
ieve
t
h
e
s
a
m
e
numbe
r
of
v
o
lta
ge
le
v
e
l
s w
i
t
h
ou
t n
o
e
xtra
c
lam
p
i
n
g
dio
d
e
s
or
vo
l
t
a
g
e
b
al
a
n
cin
g
cap
a
c
ito
rs.
c.
S
o
ft
s
w
itc
h
i
ng
t
e
ch
n
i
q
u
es c
an
be
imple
m
e
n
te
d w
h
ic
h
reduc
e
s
sw
i
t
ch
i
ng l
o
sse
s
and
de
v
i
c
e
s
t
r
esses.
d.
S
w
itch
i
n
g
r
e
d
u
nda
nc
y for in
n
e
r vo
ltage
l
e
v
e
l
s is p
ossib
l
e b
eca
u
se t
he
pha
se
v
o
l
t
a
ge o
u
t
p
u
t
i
s
the
s
um
o
f
ea
ch br
i
dge
’s o
ut
p
u
t
.
e.
P
o
ten
t
ial o
f
sh
o
ck
i
s
reduc
ed
due
t
o t
h
e
se
pa
ra
te D
C sour
ces.
2.1.
Prin
c
i
p
l
e
of o
p
erat
i
on
c
asc
a
d
e
d
mu
ltile
vel
i
n
ver
t
er
A
r
e
lati
ve
ly
n
ew
p
ow
er
c
on
verter
s
truc
ture,
c
a
sca
d
e
d
-in
v
er
t
e
rs
wi
t
h
s
e
p
arat
e
DC
s
o
u
r
c
e
s
i
s
in
t
r
od
uc
ed
h
e
r
e.
T
his
new
c
o
nver
t
er
c
a
n
a
v
o
i
d
e
x
t
ra
c
lam
p
i
ng
d
i
o
d
e
s
o
r
vo
lt
a
g
e
ba
l
a
n
c
in
g
ca
pac
i
t
o
rs.
Ea
ch
S
D
C
i
s
a
ssoc
i
ated
w
i
t
h
a
si
ng
le-p
ha
se
f
u
ll
bri
dge
i
nver
t
e
r
.
The
A
C
term
inal
v
o
lta
ge
s
of
d
i
f
fere
n
t
l
eve
l
in
verter
s
ar
e
con
n
ec
t
e
d
i
n
s
e
r
ies.
T
he
p
hase ou
t
pu
t
vo
lta
ge
i
s
sy
nt
he
si
z
e
d
b
y
th
e
su
m
o
f
f
o
u
r
i
nv
e
r
t
e
r
ou
t
put
s.
Eac
h
s
in
g
l
e-p
h
a
se
f
u
l
l
br
id
ge
i
n
v
e
r
ter
c
a
n
gene
ra
te
t
hre
e
l
eve
l
o
u
t
pu
t
s
,
+
V
dc
,
0,
a
n
d
-V
dc.
Thi
s
i
s
ma
de
p
o
s
s
i
bl
e
b
y
c
on
n
ecti
ng
th
e
D
C
s
ou
rc
e
s
s
eq
ue
n
t
i
a
ll
y
to
t
h
e
AC
si
de
v
ia
t
h
e
f
our
s
e
m
ic
o
n
duc
t
o
r p
o
w
e
r
d
e
vic
e
s
.
Eac
h
l
e
v
e
l
o
f
the
ful
l
b
ri
dge
c
on
ve
rter
c
on
sis
t
s
of
f
o
u
r
s
w
itc
h
es.
U
s
i
n
g
the
t
o
p
le
ve
l
as
t
he
e
xam
p
le
,
by
tur
n
in
g
O
N
S
1
a
n
d
S
4,
y
ie
ld
s
V
1
=
+
V
d
c
.
B
y
T
u
ri
ng
O
N
S
2
an
d
S
3
,
y
i
e
l
d
s
V
1
=
-
V
d
c
.
T
u
r
n
i
n
g
O
F
F
a
l
l
sw
it
c
h
es
y
i
e
lds
V
d
c
=
0.
S
im
il
a
r
l
y
,
the
A
C
o
ut
p
u
t
v
o
l
t
a
g
e
a
t
e
a
c
h
l
ev
e
l
can
b
e
ob
t
a
in
ed
i
n
th
e
sa
me
m
a
n
n
e
r.
Mi
ni
mu
m
h
a
rmo
n
i
c
di
sto
r
tio
n
c
a
n
b
e
o
bt
ai
n
e
d
by
c
o
n
t
r
ol
lin
g
th
e
c
ondu
c
tin
g
a
ngl
e
s
o
f
swi
t
c
h
e
s
at
d
if
f
e
ren
t
in
verter
le
v
els.
S
i
ng
le
P
ha
se
F
i
v
e
Le
vel
CM
LI
T
o
pol
og
y
I
n
f
ive
le
ve
l
ca
s
cade
d
m
u
l
t
i
l
e
ve
l
i
n
v
e
rt
e
r
,
fo
ur
s
ep
a
r
at
e
DC
s
o
u
rces
(n-1)
a
r
e
use
d
.
Th
us,
f
o
ur
f
ul
l
bri
dge
i
n
v
ert
e
r
s
a
re
c
on
ne
c
t
ed
i
n
s
e
r
i
e
s
to
o
bt
a
i
n
t
h
e
fi
ve
l
e
v
e
l
s
of
o
u
t
put
a
s
0
,
V
D
C
,
2V
D
C
,
3V
D
C
a
n
d
4
V
D
C.
T
he
H
b
ri
dge
s
a
r
e
na
me
d
as
A
,
B,
C
a
n
d
D
.
T
he
s
w
i
tch
i
ng
pa
tter
n
o
f
t
h
e
pow
er
s
w
itc
he
s
in
e
ac
h
H
b
r
id
g
e
i
s
sa
me
a
s
de
sc
ri
be
d
for
the
si
ng
le
pha
se
t
hree
l
e
v
el
c
a
s
c
a
d
ed
i
nve
r
t
e
r
exc
e
p
t
t
he
s
w
i
t
c
he
s ar
e pr
ogressed
u
p
to fo
ur
brid
g
es
f
rom
brid
ges
A to D.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Star
t
i
n
g
t
o
rque
and tor
q
ue ri
p
p
le re
duc
t
i
o
n
u
s
i
n
g SV
PWM b
a
se
d
Ve
cto
r
Con
t
r
o
l…
(S
has
hib
h
u
s
h
a
n
)
1
125
F
i
gure
1.
Casc
a
d
ed m
ul
ti
leve
l
in
v
e
rter
w
it
h 9-le
ve
l
o
u
t
p
u
t
3.
SVPWM
CONTRO
L OF
N
I
N
E
L
E
V
EL
I
NVERTER
The
ca
lc
ul
a
t
i
o
ns
f
or
t
he
7
l
e
v
e
l
i
n
v
er
t
e
r
is
a
s
d
i
sc
usse
d
in
[
8
]
an
d
ca
l
c
ul
atio
n
is
e
x
t
e
n
d
e
d
to
9
l
eve
l
in
verter
i
n
t
h
i
s
p
ape
r
a
nd
t
h
e
re
sul
t
s
are
o
b
ta
i
n
ed
.
T
h
e
h
e
x
ag
o
n
al
v
ec
tor
s
can
b
e
d
i
v
i
de
d
in
to
s
i
x
m
ajor
tria
ng
u
l
ar
s
e
c
t
o
rs
(I
to
V
I
)
.
O
n
ly
t
he
f
irs
t
s
e
c
t
o
r
of
t
he
c
oor
di
na
te
i
s
us
ed
b
eca
use
t
h
e
vec
t
or
s
l
o
c
a
t
e
d
i
n
th
e
ot
her se
ct
ors can be t
r
a
n
sform
e
d
t
o
first se
c
t
or b
y cl
oc
kw
ise r
ota
t
i
n
g b
y
a
n
an
gle
of k
*
(
pi/3) k
=
(
1
,2,3,
4
,5 f
or
sect
or
2
t
o
6).
A
s
a
l
l
t
he
s
ec
t
o
rs
a
re
i
de
nt
ic
al,
on
l
y
d
e
t
a
ils
of
s
ec
tor
I
is
g
i
v
e
n
i
n
F
i
g
u
r
e
.4.
U
s
ually,
a
(M+
1
)-
leve
l
i
nver
t
e
r
i
s
d
i
scu
ssed
h
e
re
a
s
show
n
in
F
igure.
4.
B
y
de
com
p
o
si
ng
V
ref
i
n
t
o
m
a
n
d
n
a
x
i
s
i
t
i
s
e
a
s
y
t
o
ob
ta
in t
he m
a
nd n
ax
is com
p
one
n
t
of V
ref
as V
rm
and V
rn
as
g
i
v
en
n
ext
:
V
r
m
=
(
2*
M
*
V
r
ef/3V
d
c)
sin(
π/3-
θ)
(
1
)
V
r
n =(2*
M*V
r
ef /3V
dc)
sin(
θ)
(
2
)
Whe
r
e
i
s
s
pe
e
d
o
f rota
ti
ng r
e
fe
renc
e ve
ct
or
?
A
s
t
he num
ber
o
f
le
v
els
is i
nc
r
ease
d
, the
nu
m
b
er of t
r
i
a
n
g
l
e
i
n
creases in this
way:
N
T
= 6(N
-
1)
2
Fo
r
exa
m
ple
a
7-
le
vel
in
ve
rter,
tota
Jl
n
umbe
r
of
t
ria
n
gle
i
s
2
1
6
,
b
u
t
t
h
e
s
e
l
e
c
t
i
o
n
o
f
t
r
i
a
n
g
l
e
b
y
t
h
e
p
r
o
p
o
s
e
d
me
tho
d
i
s
ve
ry
s
imple
a
nd
ge
ner
a
l
i
zed.
A
n
y
spac
e
vJec
t
o
r
l
o
c
a
t
e
d
i
n
a
ny
se
ct
or
a
nd
in
a
ny
tria
ng
le
c
an
J
be
ca
l
c
u
l
a
t
e
d
easi
l
y
from
a
nd
t
h
e
val
u
e
V
r
m
a
nd
V
r
n
e
x
p
l
a
i
n
e
d
be
l
o
w
.
A
ssum
i
n
g
V
r
e
f
a
n
d
s
h
oul
d
be
s
u
J
c
h
t
h
a
t
it
l
i
es
i
n
t
h
e
r
ecta
ngu
lar
a
r
ea
s
pe
c
i
fie
d
b
y
D
E
G
F
s
ho
w
n
i
n
F
i
gu
re.
4.
A
fter
c
a
l
c
u
l
a
ting
V
rm
an
d
V
rn
,
ca
Jlc
u
l
a
t
e
t
he
l
ow
e
r
r
ounde
d i
n
te
ge
r va
lue
(m
a
nd n)
a
s show
n
b
e
l
ow
: S
a
y,
V
rm
= 2.
6 & V
rn
= 1.85.
A
s
s
u
m
i
n
g
m
=
i
n
t
(
2
.
6
)
=
2
&
n
=
i
n
t
(
1
.
8
5
)
=
1
T
h
e
s
e
m
a
n
d
n
a
r
e
d
e
f
i
n
e
d
b
y
v
e
c
t
o
r
(
m
,
n
)
i
n
m
-
n
a
x
i
s
.
I
f
(
V
rm
+
V
rn
)
<=
(
m+n
+
1) the
n
V
r
ef loca
t
e
d
i
n t
h
e
le
ft b
o
t
t
o
m
trian
g
le D
E
F
,
o
t
h
e
r
w
i
s
e
t
h
e
t
r
i
angl
e
E
F
G
.
D
w
e
lli
n
g
Time
Calc
ula
t
io
n
The
dw
el
l
i
ng
t
ime
c
a
lc
u
l
at
i
o
n
for
eac
h
sw
i
t
chi
n
g
s
t
a
t
e
is
v
e
r
y
si
mp
l
e
a
nd
g
e
n
e
ral
i
zed
.
Su
ppo
se
a
t
a
n
y
in
st
a
n
t
the
V
r
ef
l
oc
a
t
e
d
i
n
the
Tr
ian
g
le
G
EF
s
h
o
w
n
i
n
F
i
gure
.
4.
T
he
c
or
re
sp
o
n
d
i
n
g
t
h
ree
nea
r
est
spac
e
ve
c
t
ors
a
r
e
(m
1
,
n1),
(m
2
,
n
2
)
a
n
d
(
m
3
,
n3
)
.
A
c
c
ord
i
n
g
t
o
v
o
lta
ge
-tim
e
ba
lan
c
e
e
qua
t
i
o
n
i
t
ca
n
be
s
ol
ve
t
h
r
ee
d
w
e
ll
ing
ti
m
e
T
1
,
T
2
,
a
nd T
3
as foll
o
ws:
n
1
*T
1
+n
2
*T
2
+n
3
*T
3
(3
)
m
1
*T
1
+m
2
*T
2
+m
3
*T
3
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
In
t
J Po
w El
ec
&
Dri
S
y
s
t
,
Vo
l
. 1
0
,
No
.
2
, Ju
n
e
2
019
:
1
123 –
1
1
32
1
126
Whe
r
e
T
PW
M
i
s
P
W
M
t
i
m
e
p
e
r
i
o
d
?
The
me
t
h
od
de
m
onstra
t
ed h
o
w
s
i
m
pl
e it is
t
o
determ
i
n
e th
e
trian
gle
tha
t
r
efere
n
ce
vo
l
t
a
g
e
falls
i
n an
d
to
ca
l
c
u
l
a
t
e t
h
e
d
w
ell t
i
me
s. More
im
portan
t
l
y
,
the
alg
o
ri
t
h
m
has
ano
t
her few feat
ur
es:
a.
It
i
s
a
gene
ral
alg
o
ri
t
h
m,
w
hic
h
c
an
b
e
u
s
ed
f
or
i
nve
rters
wit
h
a
n
y
n
um
b
e
r
of
v
o
l
t
a
ge
l
e
v
el
s
up
to
s
e
v
e
n
leve
l a
n
d e
v
e
n
t
u
al
ly
can
e
x
t
en
d
bey
o
nd se
ve
n leve
l
w
ith
s
om
e m
o
difi
c
a
t
ion
of
i
n
v
e
r
ter
pow
er
c
ircuit
;
b.
Th
e
r
e
are
on
ly
t
wo
s
e
t
s of
e
q
u
a
t
io
n
s
f
o
r
d
we
ll
t
i
m
e
cal
c
u
l
a
t
i
o
n
. Com
pare
d w
i
t
h
the
Ca
r
tes
i
an
co
o
rd
i
n
at
e
syste
m
w
here
t
her
e
a
re m
any se
t
s
o
f di
ffere
n
t
equa
t
i
o
n
s for
the
C
alc
u
la
ti
on
of the
dw
e
ll t
i
m
e
.
c.
The
pro
p
o
s
ed
a
lg
ori
t
h
m
co
ns
ider
s
o
v
er
-m
odul
a
tio
n
c
a
s
e
a
l
so.
It
i
s
e
a
s
y
t
o
j
u
d
g
e
w
h
e
t
h
e
r
S
V
P
W
M
i
s
i
n
over-
m
odu
l
a
t
i
on
reg
i
o
n
or
no
t
,
sim
ply
by ch
e
c
ki
n
g
the
f
o
llow
i
n
g i
n
e
qua
li
ty
:
if (
V
rm
+ V
rn
) > M,
it bec
o
m
e
s
over-
modu
l
a
ti
o
n
.
A
use
f
u
l
m
eth
od t
o
ha
n
d
l
e
t
h
i
s
s
i
t
ua
t
i
o
n
is t
o
m
ulti
pl
y t
h
e
ori
gina
l
vec
t
or
s
ay V
rm
and
V
rn
b
y
a
f
a
c
t
o
r
M
/
(V
rm
+
V
rn
)
a
n
d
t
h
e c
o
n
s
eq
uen
t
st
e
p
s
a
re
s
ame as
d
i
s
c
u
ssed
.
Figure
2.
The
v
ec
tor
selec
t
i
o
n for the
t
i
me
ca
l
c
u
la
te
d
The
in
duc
tio
n
m
o
tors
a
re
v
e
r
y
com
m
on
be
cause
t
he
y
a
r
e
ine
x
pe
ns
i
ve
a
nd
ro
b
u
st,
fin
d
i
ng
use
i
n
eve
r
y
t
hi
ng
fr
o
m
i
nd
us
t
r
ia
l
ap
pl
i
c
a
t
io
ns
s
uc
h
as
pump
s
,
fan
s
,
an
d
b
l
ow
ers
to
h
om
e
app
l
i
a
nce
s
.
Tr
adi
tio
nal
l
y,
in
duc
t
i
on
m
ot
ors
ha
ve
b
ee
n
run
a
t
a
s
i
n
g
l
e
s
p
e
e
d,
w
h
i
c
h
w
as
d
et
erm
i
ned
b
y
t
h
e
f
re
que
nc
y
of
t
he
m
ain
vo
lta
ge
a
nd
th
e
num
b
e
r
of
p
ole
s
i
n
t
h
e
m
o
t
o
r.
C
o
n
tr
ol
li
ng
the
s
pe
e
d
o
f
a
n
i
nd
uc
ti
o
n
m
otor
i
s
far
more
di
ffic
ul
t
t
h
a
n
c
on
tro
l
lin
g
the
spee
d
of
a
D
C
m
o
t
o
r
si
nc
e
the
r
e
is
n
o
l
i
ne
ar
r
e
l
a
tio
ns
hi
p
be
t
w
e
e
n
t
h
e
motor
curr
ent
a
n
d
t
h
e
resul
t
i
n
g
t
or
q
u
e
a
s
t
h
er
e
is for
a DC
mo
tor.
T
h
e
te
chni
qu
e
cal
l
e
d
v
ect
o
r
c
on
t
r
o
l
c
a
n
b
e
u
s
e
d
t
o
vary
t
he
s
peed
o
f
an
i
nd
uc
ti
on
mot
o
r
ov
e
r
a
w
i
d
e
rang
e
.
I
n
the
vec
t
or
c
o
n
tr
ol
s
c
h
em
e,
a
c
omple
x
c
urre
nt
i
s
syn
t
he
siz
e
d fro
m
t
w
o
qua
dra
t
ure
c
o
mpo
n
e
n
ts,
one
o
f
w
h
i
c
h
i
s
r
es
pon
si
bl
e
fo
r
th
e
fl
ux
l
ev
e
l
i
n
t
h
e
mo
to
r,
a
n
d
ano
t
her,
w
hic
h
c
ontr
o
ls
t
he
t
orq
u
e
prod
uc
tion
i
n
t
he
m
o
t
o
r
.
Ess
en
ti
a
lly
,
th
e
co
nt
rol
p
r
ob
l
e
m
i
s
r
e
f
o
r
mu
l
a
t
e
d
t
o
r
ese
m
b
l
e
the
co
nt
rol
of
a
D
C
mo
to
r.
V
e
c
t
o
r
c
o
nt
rol
o
f
f
e
rs
a
numbe
r
of
b
ene
f
its
i
nc
l
u
d
i
n
g
s
pee
d
c
ontro
l
over
a
w
i
de
r
ange,
prec
i
s
e
s
p
ee
d
reg
u
l
at
i
o
n,
f
ast
d
y
n
am
ic
r
espo
n
s
e,
a
nd
o
p
e
r
at
io
n
a
b
ove
b
ase
spee
d.
T
h
e
vec
t
or
c
on
tro
l
a
lgor
i
t
hm
i
s
ba
se
d
on
t
w
o
f
u
n
d
am
enta
l
ide
a
s.
T
h
e
f
irst
i
s
t
h
e
f
l
ux
a
n
d
tor
que
p
ro
du
c
i
n
g
curr
ents. A
n
in
duc
t
i
o
n
m
o
t
or
c
a
n
b
e
m
ode
le
d mos
t
s
im
ply (and c
o
n
t
ro
l
l
ed
m
ost
s
i
mp
ly) us
i
n
g t
w
o qua
d
r
a
t
ure
curr
ents
r
at
her
tha
n
t
he
f
a
m
i
l
i
a
r
t
hree
pha
se
c
urre
nts
a
c
t
ua
l
l
y
a
p
p
l
i
e
d
t
o
t
h
e
m
o
t
o
r
.
T
h
e
s
e
t
w
o
c
u
r
r
e
n
t
s
c
a
l
l
e
d
direc
t
(Id)
and
qua
dra
t
ur
e
(Iq)
a
re
r
espo
nsible
f
or
p
ro
duc
in
g
f
l
u
x
an
d
tor
que
r
espect
i
v
el
y
i
n
t
he
m
o
t
o
r
.
By
defi
ni
t
i
o
n
,
the
Iq c
urre
nt
i
s
in
pha
se
w
ith
t
he
s
ta
tor
fl
u
x
, a
n
d
Id
i
s
at r
i
ght
a
ng
les.
O
f
c
ourse,
the
ac
t
u
a
l
v
ol
ta
ge
s
app
lie
d
t
o
t
h
e
m
otor
a
nd
t
h
e
re
s
ult
i
n
g
c
urre
nts
a
r
e
in
t
he
f
a
m
il
i
a
r
three
-
p
h
ase
sy
stem
.
The
m
o
v
e
b
etwee
n
a
s
ta
tio
na
ry
r
efere
n
c
e
f
r
a
m
e
a
nd
a
re
fe
renc
e
fra
m
e
,
w
hich
i
s
r
o
t
a
t
i
n
g
syn
c
hro
n
o
u
s
w
it
h
t
h
e
sta
t
or
f
lux,
b
ec
ome
s
t
he
n
t
h
e
pr
oble
m
.
Thi
s
l
ead
s
to
t
h
e
s
econ
d
f
und
amen
t
a
l
i
d
ea
beh
i
nd ve
c
t
or c
on
t
r
o
l
.
The
se
c
o
n
d
f
unda
me
nt
a
l
idea
is
t
ha
t
of
r
e
f
ere
n
ce
f
ra
me
s.
T
h
e
i
d
e
a
o
f
a
r
ef
e
r
en
c
e
f
ra
me
is
t
o
tra
n
sf
orm
a
q
u
a
n
t
i
t
y
t
h
a
t
i
s
s
i
nus
o
i
da
l
in
o
ne
r
e
f
er
enc
e
fra
m
e
,
t
o
a
c
ons
t
a
n
t
v
a
l
ue
i
n
a
re
fer
e
nce
fra
me,
w
h
ic
h
i
s
r
o
t
at
i
ng
a
t
t
he
s
am
e
fre
que
ncy.
O
nc
e
a
sinus
o
i
d
a
l
q
u
a
n
t
i
t
y
i
s
t
r
ansfor
me
d
to
a
c
ons
ta
nt
v
a
l
ue
b
y
c
a
re
fu
l
choi
c
e
o
f
r
e
f
ere
n
ce
f
ra
me,
i
t
b
eco
me
s
po
ssib
l
e
t
o
c
o
n
t
r
o
l
t
ha
t
qu
ant
ity
w
i
t
h
tr
adi
tio
na
l
pr
op
or
t
i
o
n
a
l
in
t
e
gral (
P
I
) c
ont
r
o
l
l
er
s.
S
p
ac
e
vec
t
or
m
odu
la
ti
o
n
(
S
V
M)
i
s
based
on
ve
ct
or
s
ele
c
tio
n
i
n
t
h
e
q
-
d
s
t
a
t
i
o
n
a
r
y
r
e
f
e
r
e
n
c
e
f
r
a
m
e
.
The
c
o
mm
and
e
d
v
o
l
ta
ge
v
ec
tor
is
d
e
f
ine
d
by
e
q
uat
i
on-
1.t
o
4
.
Th
e
c
om
ma
nd
e
d
v
ec
tor
is
p
lo
tte
d
a
l
ong
w
i
t
h
the
vec
t
or
s
o
b
t
a
ina
b
le
by
t
h
e
in
ver
t
e
r
.
The
d
e
sired
ve
c
t
or
V
s*
qds
s
how
n
at
s
om
e
p
o
i
n
t
in
tim
e,
b
ut
w
i
l
l
fo
llo
w
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Star
t
i
n
g
t
o
rque
and tor
q
ue ri
p
p
le re
duc
t
i
o
n
u
s
i
n
g SV
PWM b
a
se
d
Ve
cto
r
Con
t
r
o
l…
(S
has
hib
h
u
s
h
a
n
)
1
127
the
c
i
rcu
l
ar
p
a
t
h
i
f
a
t
hree
p
ha
se
s
e
t
o
f
v
o
lta
g
e
s
a
r
e
re
qui
r
e
d
on
t
h
e
l
o
a
d
.
The
first
ste
p
i
n
t
h
e
S
V
M
sc
he
me
i
s
to i
de
nt
ify t
h
e
thr
ee
nea
r
est ve
c
t
ors.
F
i
gure
3.
H
exa
gona
l
S
V
P
W
M
e
x
ten
s
io
n
for
9 leve
l
t
h
r
ee-
p
h
ase
inv
ert
e
r
4.
RESULT
S
A
N
D
ANALY
S
IS
The
ve
c
t
or c
on
t
r
o
l
of in
duc
t
i
o
n m
a
c
h
ine
is d
one
in s
i
mula
t
i
on o
f
MATLA
B. Two
c
ases are
a
naly
zed
one i
s
w
ith tw
o
l
eve
l
i
n
v
erte
r
s
.
The
SV
P
W
M
is
u
se
d
he
re
a
s
the
P
W
M
m
odu
la
tio
n.
A
no
t
h
er
c
ase
i
s
w
it
h
n
i
ne
leve
l
ca
sc
a
d
e
d
m
ulti
le
vel
in
ve
rter
w
i
t
h
S
V
P
W
M
base
d
vec
t
or
c
o
n
t
r
o
l
.
A
nd
i
n
n
i
n
e
l
e
vel
i
n
ve
rters
i
s
f
ed
w
it
h
so
l
a
r
ba
sed
bo
os
t
co
nve
rter
w
ith
i
nc
rem
e
nt
al
c
o
n
d
u
c
t
a
n
ce
a
lg
ori
t
h
m.
B
ot
h
t
h
e
ca
se
s
a
r
e
a
n
al
y
zed
w
i
t
h
10
0
rad/sec
as
r
efere
n
ce
spee
d
an
d
l
o
a
d
t
orq
u
e
i
s
50
Nm
from
0
to
1
.5
s
ecs.
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ig
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Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
In
t
J Po
w El
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y
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l
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0
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2
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019
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32
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128
F
i
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S
pee
d
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o
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g
raph
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ve
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l
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c
k
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eed i
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m
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
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c
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D
ri S
yst
IS
S
N
:
2088-
86
94
Star
t
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g
t
o
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le re
duc
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se
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1
129
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i
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i
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Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
In
t
J Po
w El
ec
&
Dri
S
y
s
t
,
Vo
l
. 1
0
,
No
.
2
, Ju
n
e
2
019
:
1
123 –
1
1
32
1
130
F
i
gure
8
. Nin
e-lev
e
l i
nvert
er with
vect
or co
n
t
r
o
l
THD an
a
l
ysis
The TH
D
va
lu
e
of tw
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n
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r
ter cur
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ent
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n
d
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sh
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s
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a
bl
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P
e
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rmance co
mp
aris
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Se
ttli
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n
s
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R
ise
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s
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r
ting torque
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e
2
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e
pic
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ane
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l
e 2
.
S
o
l
ar p
o
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p
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fi
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tions
PV
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n
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e
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PR
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rov
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ab
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e
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ase
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ontent
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y
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F
T
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300
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40
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80
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r
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ag
(
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o
f
F
unda
mental
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Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Star
t
i
n
g
t
o
rque
and tor
q
ue ri
p
p
le re
duc
t
i
o
n
u
s
i
n
g SV
PWM b
a
se
d
Ve
cto
r
Con
t
r
o
l…
(S
has
hib
h
u
s
h
a
n
)
1
131
T
a
ble
3. M
oto
r
p
aram
eters
Pa
r
a
m
e
te
rs
S
pec
i
fi
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a
tion
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we
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0
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l R
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S Vol
t
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que
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c
y
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z
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er
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f
p
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es
2
5.
CONCL
U
S
ION
The
v
e
c
t
or
c
ontr
o
l
o
f
i
n
duc
tio
n
mot
o
r
is
f
ed
w
it
h
tw
o-le
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e
l
in
v
e
r
t
e
r
a
n
d
ni
n
e
l
ev
el
i
nv
ert
e
rs
a
nd
SV
P
W
M
t
ech
ni
qu
e
i
s
u
se
d
c
o
n
t
ro
l
it
.
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e
ni
n
e
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ev
e
l
i
n
v
e
rte
r
i
s
fe
d
wi
th
s
o
l
ar
a
nd
b
o
o
s
t
co
nv
e
r
t
e
r
as
i
np
ut
a
s
the
num
ber
o
f
D
C
sour
ces
a
re
h
ig
h.
T
he
p
er
form
anc
e
p
a
r
am
eter
s
l
ike
se
ttl
i
ng
t
i
m
e
,
rise
time
,
T
H
D
a
nd
st
a
r
t
i
n
g
t
orq
u
e
s
a
re
r
educ
ed
i
n
pro
pose
d
n
in
e
l
e
vel s
o
lar
-
fe
d
S
V
P
W
M
i
nvert
e
r
.
REFE
RENCES
[1]
N
a
u
m
an
en
J
.
K
o
rh
on
en
P
.
Silv
e
n
t
o
inen
J
.
P
y
rh
o
̈
ne
n
,
“
Mit
i
gatio
n
o
f
h
i
gh
du
/dt-o
r
igi
n
at
ed
m
otor
o
v
e
rv
oltag
e
s
in
m
u
lti
l
evel in
v
erter dri
v
es
”,
IET Power
El
ectr
o
n
i
cs
,
20
09,
P
age
N
o
.
6
8
1
-
6
89.
[2]
A
.
C
h
i
t
r
a
,
S
.
H
i
m
a
v
a
t
h
i
,
“
R
e
d
u
c
e
d
swi
tch
mul
til
e
v
el
i
nver
t
e
r
fo
r
pe
r
f
orma
n
c
e
e
n
h
a
nce
m
e
n
t
o
f
i
nd
uc
tion
mo
to
r
dri
v
e
wit
h
i
nt
elligen
t
r
o
t
o
r
resist
a
nce es
timat
or”,
IET Po
wer
Elect
ro
nics
,
2
0
1
5
,
pp.
1-10
.
[3]
S
é
rgi
o
D
ah
er,
Jü
rgen
S
ch
m
i
d
,
a
nd
F
e
rn
ando
L
.
M
.
A
nt
un
e
s
,
“
M
ul
ti
le
ve
l
In
ve
rte
r
T
op
o
l
og
ie
s
fo
r
Sta
n
d
-
Alon
e
PV
Sy
ste
m
s”
,
IEE
E
T
r
ans
actio
ns
on
Ind
u
s
t
ria
l
El
ectron
i
cs
,
Vo
l. 5
5,
N
o. 7
,
Ju
l
y
20
0
8
.
[4]
Jey
r
aj
S
elvaraj
a
n
d
Nasru
d
i
n
A
.
Rahi
m
,
“
M
u
ltil
e
vel
Inv
e
rter
f
or
G
rid
-
Con
n
ected
P
V
S
y
stem
E
m
p
lo
yin
g
D
i
g
i
t
a
l
P
I
Controller”,
IEEE
T
r
a
n
s
a
c
t
io
n
s
on
Ind
u
stri
al El
e
c
tr
on
ics
,
Vol
.
5
6
, No
. 1
,
Jan
u
a
ry 2
00
9.
[5]
Ananda
rup
Das
,
K
.
Sivakum
a
r,
R
i
j
il
Ramchand,
Chin
tan
Pa
t
e
l
a
n
d
K
.
G
o
p
aku
m
ar,
“A
C
om
bi
nation
o
f
H
exag
on
al
and
12
-S
id
ed
P
oly
gon
al Vo
ltag
e
S
pace Vecto
r
P
WM
Co
n
t
r
o
l
f
or IM
Dri
v
es U
s
i
ng
Cas
caded T
w
o
-Lev
el
I
n
v
ert
e
rs”,
IEEE
Transac
t
i
ons On In
dustrial E
l
ectronics
, Vo
l
. 5
6,
N
o.
5,
May
2
0
0
9
.
[6]
K.
S
i
v
aku
m
ar,
An
andaru
p
D
a
s
,
R
ijil
Ram
c
h
a
nd,
C
h
i
ntan
P
at
el
a
nd
K.
G
opakumar,
“A
H
ybr
i
d
M
ul
tilevel
Inverter
To
po
log
y
f
or
a
n
Op
en-E
nd
Wi
nding
I
n
duct
i
o
n
-Mo
t
or Driv
e
U
sin
g
Tw
o
-
Lev
e
l
In
ve
rt
ers
i
n
S
eri
e
s
W
i
t
h
a
Capacit
o
r-
Fe
d
H-Brid
ge
C
e
l
l
”
,
I
E
EE T
r
ans
a
cti
o
ns
On
Indust
r
ial E
l
ect
roni
c
s
,
V
o
l.
5
7
,
N
o.
1
1
,
N
o
v
e
m
b
e
r
2010
.
[7]
Jav
i
er
C
hav
a
rría,
D
omin
go
Bi
el,
F
r
ances
c
Gui
n
joan
,
Carlos
M
eza
,
a
nd
J
u
a
n
J.
N
e
g
ro
ni,
“Ene
rg
y-
Ba
la
n
c
e
Co
ntrol
of
P
V
Cascaded
M
ultilevel
Gr
id-Con
nected
I
nve
r
t
e
rs
u
nder
L
evel-
Shi
f
ted
a
nd
Phase-Shifted
P
W
M
s
”,
IE
EE
Tran
sac
t
io
ns
on
Ind
u
str
i
a
l
Ele
c
tr
on
ic
s
,
Vo
l
.
6
0,
N
o
.
1
,
Jan
u
ary
20
13.
[8]
K.C
Jana
a
nd
S
ujit
Kumar
Biswas,"A
s
impl
e
and
Generali
ze
d
SVP
W
M
C
on
trol
o
f
Cascad
ed
H
-Bri
dge
M
ulti
level
Inverters”,
Jou
r
na
l o
f
El
ectri
ca
l
En
gi
neeri
n
g
.
[9]
Anubra
t
a
D
ey,
P.
P
.
Rajeevan,
R
ijil
R
amchand,
K
.
Ma
thew,
and
K.
G
opak
u
m
a
r,
“
A
S
p
ace-V
ecto
r
-Bas
ed
Hyst
e
r
esis
C
urre
nt
C
on
t
r
oller
f
o
r
a
G
e
neral
n-L
e
v
e
l
Inve
rt
er-Fe
d
D
r
iv
e
With
N
earl
y
Co
n
stan
t
Sw
itchi
ng F
r
equen
c
y
Control
”
,
IEEE
T
r
a
n
sa
cti
o
n
s
O
n
In
du
str
i
a
l
Electro
ni
cs
,
Vo
l
.
60
,
No
.
5
, May
20
1
3
.
[10]
Am
arend
r
a
E
d
p
u
g
a
nti
&
Aksh
ay
K
.
Rath
ore,
“
A
S
u
rv
ey
o
f
Low-S
w
itc
hin
g
F
re
qu
ency
M
od
ul
ati
o
n
T
e
c
h
ni
qu
es
f
or
Medi
um-
V
ol
t
a
ge Mul
tilevel Con
v
erters”,
IACC
,
20
15
pp
.
1-8
.
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